Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models
This work addresses inference from deep generative priors in applications with complex correlation structures, offering a more efficient algorithm for researchers in signal processing and machine learning, though it is incremental as it builds on existing AMP frameworks.
The authors tackled the problem of reconstructing signals and hidden variables from observations in multi-layer networks with rotationally invariant weight matrices, which generalizes beyond Gaussian assumptions. They introduced ML-RI-GAMP, a new approximate message passing algorithm that achieves significantly lower complexity than existing methods like ML-VAMP, with little to no performance loss in numerical results.
We consider the problem of reconstructing the signal and the hidden variables from observations coming from a multi-layer network with rotationally invariant weight matrices. The multi-layer structure models inference from deep generative priors, and the rotational invariance imposed on the weights generalizes the i.i.d.\ Gaussian assumption by allowing for a complex correlation structure, which is typical in applications. In this work, we present a new class of approximate message passing (AMP) algorithms and give a state evolution recursion which precisely characterizes their performance in the large system limit. In contrast with the existing multi-layer VAMP (ML-VAMP) approach, our proposed AMP -- dubbed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) -- provides a natural generalization beyond Gaussian designs, in the sense that it recovers the existing Gaussian AMP as a special case. Furthermore, ML-RI-GAMP exhibits a significantly lower complexity than ML-VAMP, as the computationally intensive singular value decomposition is replaced by an estimation of the moments of the design matrices. Finally, our numerical results show that this complexity gain comes at little to no cost in the performance of the algorithm.